Show commands:
SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 123200.gn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123200.gn1 | 123200v2 | \([0, -1, 0, -2279233, 929702337]\) | \(1278763167594532/375974556419\) | \(384997945773056000000\) | \([2]\) | \(3932160\) | \(2.6558\) | |
123200.gn2 | 123200v1 | \([0, -1, 0, 382767, 96496337]\) | \(24226243449392/29774625727\) | \(-7622304186112000000\) | \([2]\) | \(1966080\) | \(2.3092\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123200.gn have rank \(0\).
Complex multiplication
The elliptic curves in class 123200.gn do not have complex multiplication.Modular form 123200.2.a.gn
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.